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Theorem cadanOLD 1435
Description: Obsolete proof of cadan 1434 as of 25-Sep-2018. (Contributed by Mario Carneiro, 4-Sep-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
cadanOLD  |-  (cadd (
ph ,  ps ,  ch )  <->  ( ( ph  \/  ps )  /\  ( ph  \/  ch )  /\  ( ps  \/  ch ) ) )

Proof of Theorem cadanOLD
StepHypRef Expression
1 ordi 859 . . . 4  |-  ( ( ( ( ph  /\  ps )  \/  ( ph  /\  ch ) )  \/  ( ps  /\  ch ) )  <->  ( (
( ( ph  /\  ps )  \/  ( ph  /\  ch ) )  \/  ps )  /\  ( ( ( ph  /\ 
ps )  \/  ( ph  /\  ch ) )  \/  ch ) ) )
2 ordir 860 . . . . . 6  |-  ( ( ( ph  /\  ch )  \/  ps )  <->  ( ( ph  \/  ps )  /\  ( ch  \/  ps ) ) )
3 simpr 461 . . . . . . . . . . 11  |-  ( (
ph  /\  ps )  ->  ps )
43con3i 135 . . . . . . . . . 10  |-  ( -. 
ps  ->  -.  ( ph  /\ 
ps ) )
5 biorf 405 . . . . . . . . . 10  |-  ( -.  ( ph  /\  ps )  ->  ( ( ph  /\ 
ch )  <->  ( ( ph  /\  ps )  \/  ( ph  /\  ch ) ) ) )
64, 5syl 16 . . . . . . . . 9  |-  ( -. 
ps  ->  ( ( ph  /\ 
ch )  <->  ( ( ph  /\  ps )  \/  ( ph  /\  ch ) ) ) )
76pm5.74i 245 . . . . . . . 8  |-  ( ( -.  ps  ->  ( ph  /\  ch ) )  <-> 
( -.  ps  ->  ( ( ph  /\  ps )  \/  ( ph  /\ 
ch ) ) ) )
8 df-or 370 . . . . . . . 8  |-  ( ( ps  \/  ( ph  /\ 
ch ) )  <->  ( -.  ps  ->  ( ph  /\  ch ) ) )
9 df-or 370 . . . . . . . 8  |-  ( ( ps  \/  ( (
ph  /\  ps )  \/  ( ph  /\  ch ) ) )  <->  ( -.  ps  ->  ( ( ph  /\ 
ps )  \/  ( ph  /\  ch ) ) ) )
107, 8, 93bitr4i 277 . . . . . . 7  |-  ( ( ps  \/  ( ph  /\ 
ch ) )  <->  ( ps  \/  ( ( ph  /\  ps )  \/  ( ph  /\  ch ) ) ) )
11 orcom 387 . . . . . . 7  |-  ( ( ( ph  /\  ch )  \/  ps )  <->  ( ps  \/  ( ph  /\ 
ch ) ) )
12 orcom 387 . . . . . . 7  |-  ( ( ( ( ph  /\  ps )  \/  ( ph  /\  ch ) )  \/  ps )  <->  ( ps  \/  ( ( ph  /\  ps )  \/  ( ph  /\  ch ) ) ) )
1310, 11, 123bitr4i 277 . . . . . 6  |-  ( ( ( ph  /\  ch )  \/  ps )  <->  ( ( ( ph  /\  ps )  \/  ( ph  /\  ch ) )  \/  ps ) )
14 orcom 387 . . . . . . 7  |-  ( ( ch  \/  ps )  <->  ( ps  \/  ch )
)
1514anbi2i 694 . . . . . 6  |-  ( ( ( ph  \/  ps )  /\  ( ch  \/  ps ) )  <->  ( ( ph  \/  ps )  /\  ( ps  \/  ch ) ) )
162, 13, 153bitr3i 275 . . . . 5  |-  ( ( ( ( ph  /\  ps )  \/  ( ph  /\  ch ) )  \/  ps )  <->  ( ( ph  \/  ps )  /\  ( ps  \/  ch ) ) )
17 simpr 461 . . . . . . . . . . 11  |-  ( (
ph  /\  ch )  ->  ch )
1817con3i 135 . . . . . . . . . 10  |-  ( -. 
ch  ->  -.  ( ph  /\ 
ch ) )
19 biorf 405 . . . . . . . . . . 11  |-  ( -.  ( ph  /\  ch )  ->  ( ( ph  /\ 
ps )  <->  ( ( ph  /\  ch )  \/  ( ph  /\  ps ) ) ) )
20 orcom 387 . . . . . . . . . . 11  |-  ( ( ( ph  /\  ch )  \/  ( ph  /\ 
ps ) )  <->  ( ( ph  /\  ps )  \/  ( ph  /\  ch ) ) )
2119, 20syl6bb 261 . . . . . . . . . 10  |-  ( -.  ( ph  /\  ch )  ->  ( ( ph  /\ 
ps )  <->  ( ( ph  /\  ps )  \/  ( ph  /\  ch ) ) ) )
2218, 21syl 16 . . . . . . . . 9  |-  ( -. 
ch  ->  ( ( ph  /\ 
ps )  <->  ( ( ph  /\  ps )  \/  ( ph  /\  ch ) ) ) )
2322pm5.74i 245 . . . . . . . 8  |-  ( ( -.  ch  ->  ( ph  /\  ps ) )  <-> 
( -.  ch  ->  ( ( ph  /\  ps )  \/  ( ph  /\ 
ch ) ) ) )
24 df-or 370 . . . . . . . 8  |-  ( ( ch  \/  ( ph  /\ 
ps ) )  <->  ( -.  ch  ->  ( ph  /\  ps ) ) )
25 df-or 370 . . . . . . . 8  |-  ( ( ch  \/  ( (
ph  /\  ps )  \/  ( ph  /\  ch ) ) )  <->  ( -.  ch  ->  ( ( ph  /\ 
ps )  \/  ( ph  /\  ch ) ) ) )
2623, 24, 253bitr4i 277 . . . . . . 7  |-  ( ( ch  \/  ( ph  /\ 
ps ) )  <->  ( ch  \/  ( ( ph  /\  ps )  \/  ( ph  /\  ch ) ) ) )
27 orcom 387 . . . . . . 7  |-  ( ( ( ph  /\  ps )  \/  ch )  <->  ( ch  \/  ( ph  /\ 
ps ) ) )
28 orcom 387 . . . . . . 7  |-  ( ( ( ( ph  /\  ps )  \/  ( ph  /\  ch ) )  \/  ch )  <->  ( ch  \/  ( ( ph  /\  ps )  \/  ( ph  /\  ch ) ) ) )
2926, 27, 283bitr4i 277 . . . . . 6  |-  ( ( ( ph  /\  ps )  \/  ch )  <->  ( ( ( ph  /\  ps )  \/  ( ph  /\  ch ) )  \/  ch ) )
30 ordir 860 . . . . . 6  |-  ( ( ( ph  /\  ps )  \/  ch )  <->  ( ( ph  \/  ch )  /\  ( ps  \/  ch ) ) )
3129, 30bitr3i 251 . . . . 5  |-  ( ( ( ( ph  /\  ps )  \/  ( ph  /\  ch ) )  \/  ch )  <->  ( ( ph  \/  ch )  /\  ( ps  \/  ch ) ) )
3216, 31anbi12i 697 . . . 4  |-  ( ( ( ( ( ph  /\ 
ps )  \/  ( ph  /\  ch ) )  \/  ps )  /\  ( ( ( ph  /\ 
ps )  \/  ( ph  /\  ch ) )  \/  ch ) )  <-> 
( ( ( ph  \/  ps )  /\  ( ps  \/  ch ) )  /\  ( ( ph  \/  ch )  /\  ( ps  \/  ch ) ) ) )
331, 32bitri 249 . . 3  |-  ( ( ( ( ph  /\  ps )  \/  ( ph  /\  ch ) )  \/  ( ps  /\  ch ) )  <->  ( (
( ph  \/  ps )  /\  ( ps  \/  ch ) )  /\  (
( ph  \/  ch )  /\  ( ps  \/  ch ) ) ) )
34 df-3or 966 . . 3  |-  ( ( ( ph  /\  ps )  \/  ( ph  /\ 
ch )  \/  ( ps  /\  ch ) )  <-> 
( ( ( ph  /\ 
ps )  \/  ( ph  /\  ch ) )  \/  ( ps  /\  ch ) ) )
35 anandir 825 . . 3  |-  ( ( ( ( ph  \/  ps )  /\  ( ph  \/  ch ) )  /\  ( ps  \/  ch ) )  <->  ( (
( ph  \/  ps )  /\  ( ps  \/  ch ) )  /\  (
( ph  \/  ch )  /\  ( ps  \/  ch ) ) ) )
3633, 34, 353bitr4i 277 . 2  |-  ( ( ( ph  /\  ps )  \/  ( ph  /\ 
ch )  \/  ( ps  /\  ch ) )  <-> 
( ( ( ph  \/  ps )  /\  ( ph  \/  ch ) )  /\  ( ps  \/  ch ) ) )
37 cador 1433 . 2  |-  (cadd (
ph ,  ps ,  ch )  <->  ( ( ph  /\ 
ps )  \/  ( ph  /\  ch )  \/  ( ps  /\  ch ) ) )
38 df-3an 967 . 2  |-  ( ( ( ph  \/  ps )  /\  ( ph  \/  ch )  /\  ( ps  \/  ch ) )  <-> 
( ( ( ph  \/  ps )  /\  ( ph  \/  ch ) )  /\  ( ps  \/  ch ) ) )
3936, 37, 383bitr4i 277 1  |-  (cadd (
ph ,  ps ,  ch )  <->  ( ( ph  \/  ps )  /\  ( ph  \/  ch )  /\  ( ps  \/  ch ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    \/ w3o 964    /\ w3a 965  caddwcad 1421
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-xor 1352  df-cad 1423
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator